Re: Integrate 5.0

• To: mathgroup at smc.vnet.net
• Subject: [mg44242] Re: [mg44233] Integrate 5.0
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Fri, 31 Oct 2003 03:01:07 -0500 (EST)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <200310290834.DAA05970@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```I tried your integral and saw the same quite complicated "If" result.
So I did what seemed the obvious thing (with output here converted to
InputForm):

Integrate[Sqrt[Cos[t] + 1], {t, 0, x}, Assumptions -> x ? Reals]

If[x <= Pi && Pi + x >= 0, 2*Sqrt[1 + Cos[x]]* Tan[x/2],
Integrate[Sqrt[1 + Cos[t]], {t, 0, x},
Assumptions -> x ? Reals && (x > Pi || Pi + x < 0)]]

That doesn't seem so awful to me.

Selwyn Hollis wrote:
> I've come to the conclusion that Integrate has become nearly worthless
> for computing definite integrals with symbolic limits. To cite a simple
> example,
>
> 	Integrate[Sqrt[Cos[t] + 1], {t, 0, x}]
>
> returns an awful mess inside of an If statement (very mild in this
> case) that no one should have to deal with if they're only concerned
> with real numbers (specifically calculus students and a great many
> applied mathematicians).
>
> On the other hand, DSolve gives the simple, clean answer that Integrate
> used to give:
>
>     y[t]/. DSolve[{y'[t] == Sqrt[Cos[t] + 1], y[0] == 0}, y[t],t]
>
> 	   2*Sqrt[1 + Cos[t]]*Tan[t/2]
>
> Could it be that we need a new function such as this:
>
> 	RealIntegral[expr_,{x_,a_,b_}]:=
> 		(y[x]/. First@DSolve[{y'[x] ==expr, y[a] == 0}, y[t], t])/.x->b
>
> that would be associated with \[Integral] ? ... leaving the current
> Integrate to be associated with \[ContourIntegral]??
>
> Or perhaps a simple option for Integrate like RealLimits->True?
>
> -----
> Selwyn Hollis
> http://www.math.armstrong.edu/faculty/hollis
>
>

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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